Sensor arrays are known in the art for spatially sampling wave fronts at a given frequency. The most obvious application is a microphone array embedded in a telephone set, to provide conference call functionality. In order to avoid spatial sampling aliasing, the distance, d, between sensors must be lower than λ/2 where λ is the wavelength.
Many publications are available on the subject of sensor arrays, including:    [1] A. Ishimaru, “Theory of unequally spaced arrays”, IRE Trans Antenna and Propagation, vol. AP-10, pp.691-702, November 1962    [2] Jens Meyer, “Beamforming for a circular microphone array mounted on spherically shaped objects”, Journal of the Acoustical Society of America 109 (1), January 2001, pp. 185-193.    [3] Marc Anciant, “Mod{acute over (e)}lisation du champ acoustique incident au décollage de la fusée Ariane”, July 1996, Ph.D. Thesis, Université de Technologie de Compiègne, France.    [4] Michael Stinson, James Ryan, “Microphone array diffracting structure”, Canadian Patent Application 2,292,357.    [5] P. J. Kootsookos, D. B. Ward, R. C. Williamson, “Imposing pattern nulls on broadband array responses”, Journal of the Acoustical Society of America 105 (6, June 1999, pp. 3390-3398.    [6] Henry Cox, Robert Zeskind, Mark Owen, “Robust Adaptive Beamforming”, IEEE Trans. on Acoustics, Speech, and Signal Processing, Vol. ASSP-35, No. 10 October 1987, pp.1365-1376    [7] Feng Qian “Quadratically Constrained Adaptive Beamforming for Coherent Signals and Inteference”, IEEE Trans. On Signal Proc. Vol.43 No.8 August 1995, pp.1890-1900    [8] Zhi Tian, K. Bell, H. L. Van Trees “A Recursive Least Squares Implementation for LCMP Beamforming Under Quadratic Constraint”, IEEE Trans. On Signal Processing, Vol. 49, No. 6, June 2001, pp.1138-1145    [9] O. L. Frost, “An algorithm for linearly constrained adaptive array processing”, Proceedings IEEE, vol. 60, pp. 926-935, august 1972.    [10] J. Lardies, “Acoustic ring array with constant beamwidth over a very wide frequency range”, Acoustics Letters, vol. 13, pp. 77-81, November 1989.    [11] M. F. Berger and H. F. Silverman, “Microphone array optimization by stochastic region contraction”, IEEE Trans, Signal Processing”, vol. 39, pp.2377-2386, November 1991.    [12] F. Pirz, “Design of a wideband, constant beamwidth array microphone for use in the near field”, Bell Systems Technical Journal, vol. 58, pp. 1839-1850, October 1979.    [13] D. Ward, R. A. Kennedy, R. C. Williamson, “Theory and design of broadband sensor arrays with frequency invariant far-field beam-patterns”, Journal of The Acoustical Society of America, vol. 97, pp. 1023-1034, February 1995.    [14] Gary Elko, “A steerable and variable first-order differential microphone array”, U.S. Pat. No. 6,041,127, Mar. 21, 2000.    [15] M. I. Skolnik “Non uniform arrays”, in “Antenna Theory”, Pt. 1, edited by R. E. Collin and F. Jzucker (Mc GrawHill, New-York, 1969), Chap. 6, pp. 207-279    [16] A. C. C. Warnock & W. T. Chu, “Voice and Background noise levels measured in open offices”, IRC Internal Report IR-837, January 2002.    [17] Morse and Ingard, “Theoretical Acoustics”, Princeton University Press, 1968.    [18] Michael Brandstein, Darren. Ward, “Microphone arrays”, Springer, 2001.
For free-field linear, circular, or non-linear arrays, Ishimaru [1] discusses the issues of constant inter sensor spacing and non-constant inter-sensor spacing.
Meyer [2] discloses arrays embedded in a diffracting obstacle of simple shape, and provides an analytical solution for the wave equation in acoustics. For arrays of simple shape like circular rings embedded in a more complex shape, for which there is no analytical solution of the wave equation, Anciant [3] and Ryan [4] make use of numerical methods, such as Boundary Element methods (BEM) or Finite or Infinite Elements methods (FEM, IFEM).
Most of the literature describes broadband frequency invariant beamforming for circular arrays or linear arrays, but not for microphone arrays in shapes that are not symmetric or axi-symmetric. One example of such an obstacle whose shape is dictated by industrial design constraints resulting in an odd shape, is a telephone incorporating a microphone array. The problem of beamforming with such an array is quite different from that dealt with in the literature since the solution relies on constrained optimisation, with a constraint build using a set of vectors containing the sensor signal for acoustic waves with specific directions of arrival.
In that regard, the following prior art is relevant:
P. Kootssokos [5] proposes a technique intended for rejecting a far-field broadband signal from a given known direction by imposing pattern nulls on broadband array responses. The method consists of generating deep and wide “null” or quiescent areas in given directions. This is achieved by imposing a set of linear constraints.
Henry Cox [6] proposes robust adaptive beamforming by the use of different sets of constraints. The constraints, quadratic and linear, are used to make the beamformer more robust to small errors of sensor amplitude, phase or position.
Feng Qian [7] proposes a quadratically constrained adaptive beamforming technique, but deals only with coherent interfering signals.
In Zhi Tian, K Bell, H. L. Van Trees [8], LCMP beamforming is set forth under quadratic constraints to provide an adaptive beamformer, but is concerned only with the stability of convergence.
Although a number of the methods discussed in the above-referenced prior art use specific vectors to shape the beam they, do not deal with the consequences of non-linear or non axi-symmetric arrays on the beampatterns and the resultant possible loss of “look” direction.
The following prior art relates more specifically to beamforming with constant broadband frequency invariant beamwidth, but not in relation to non axi-symmetric or non-linear arrays:
Frost [9] sets forth an adaptive array with M sensors to produce M constraints on the beam pattern of the array at a single frequency. The author proposes an algorithm for linearly constrained adaptive array processing. A set of linear constraints is introduced to provide an adaptive process in order to build a super directive array. Although this method can produce a constant beam pattern or null in given directions at various frequencies it is not designed to produce an identical beam pattern over a continuous frequency band and for various azimuth angle when the array is “asymmetric”.
Lardies [10] proposes an acoustic multiple ring array with constant beamwidth over a very wide frequency range. To determine the unknown filter function, a linear constraint is imposed at an angle θH corresponding to the half-power beam angle. This procedure is intended to generate a constant beam over a band of frequencies, but is limited to symmetrical free-field arrays.
Berger and Silverman [11] disclose another approach consisting of designing the broadband sensor array by determining sensor gains and inter-sensor spacing as a multidimensional optimisation problem. This method does not use frequency dependant array sensor gains but attempts to find optimal spacing and fixed gains by minimising the array power spectral density over a given frequency band
Pirz [12] uses harmonic nesting, in which the array is composed of several sets of sub-arrays with different inter-sensor spacings adapted for different frequency ranges. It should be noted that lowering the inter-sensor spacing under λ/2 only provides redundant information and directly conflicts with the desire to have as much aperture as possible for a fixed number of sensors.
Ishimaru [1] uses the asymptotic theory of unequally spaced arrays to derive relationships between beam pattern properties (peal response, main lobe width, . . . ) and array design. These relationships are then used to translate beam pattern requirements into functional requirements on the sensor spacing and weighting, thereby deriving a constant broadband design.
The prior art culminates with Ward [13] who finds a more general solution for providing the best possible broadband frequency invariant beam pattern. Ward considers a broadband array with constant beam pattern in the far field. Again, the asymptotic theory of unequally spaced arrays is used to derive relationships between beam pattern properties such as main lobe width, peak response, and array design. These relationships are expressed versus sensor spacing and weightings and Ward uses an ideal continuous sensor that is then “discretised” in an optimal array of point sensors, giving constant broadband beamwidth.
The following prior art relates to arrays embedded in obstacles:
The benefit of an obstacle for a microphone array in terms of directivity and localisation of the source or multiple sources is discussed in Marc Anciant [4]. Anciant describes the “shadow” area induced by an obstacle for a 3D-microphone array around a mock-up of the Ariane IV launcher in detecting and characterising the engine noise sources at takeoff.
Meyer [2] uses the concept of phase mode to generate a desired beam pattern from a circular array embedded in a rigid sphere, taking advantage of the analytical expression of the pressure diffracted by such an obstacle. He describes the benefit of the obstacle in term of broadband performance and noise susceptibility improvement
Elko [14] uses a small sphere with microphone dipoles in order to increase wave-travelling time from one microphone to another and thus achieve better performance in terms of directivity. A sphere is used since it allows for analytical expressions of the pressure field generated by the source and diffracted by the obstacle. The computation of the pressure at various points on the sphere allows the computation of each microphone signal weight.
Jim Ryan et al [4] extend this idea to circular microphone arrays embedded in obstacles with more complex shapes using a super-directive approach and a boundary element method to compute the pressure field diffracted by the obstacle. Emphasis is placed on the low frequency end, to achieve strong directivity with a small obstacle and a specific impedance treatment for allowing air-coupled surface waves to occur. This treatment results in increasing the wave travel time from one microphone to another thereby increasing the “apparent” size of the obstacle for better directivity in the low frequency end. Ryan et al. have shown that using an obstacle improves directivity in the low frequency domain, compared to the same array in free field.
Skolnik [15] is noteworthy for teaching that error occurs when the position of the array sensors are subject to variation, and by extension that this error can be applied to non-uniform arrays.
Except for Anciant and Ryan, none of the techniques described in the prior art can be used when the sensor array is embedded in an obstacle with an odd shape, in the presence of a rigid plane for example, either with or without an acoustic impedance condition on its surface. Numerical methods are required. As they do not give an analytical expression of the pressure field at the sensor vs. frequency, the techniques proposed by most of the above-referenced authors (except Anciant and Ryan) can not be used. None of the prior art deals with or describes variation of the beam pattern in such conditions. It should be noted that Anciant and Ryan deal with circular arrays only, and do not deal with constant beamwidth or any other problem linked to frequency variation and array geometry properties.